Logarithms are a set of numbers that have been computed and formed into tables, for the purpose of facilitating many difficult arithmetical calculations; being so contrived, that the addition and subtraction of them answers to the multiplication and division of natural numbers with which they are made to correspondt.

* The series here treated of are such as are usually called algebraical which, of course, embrace only a small part of the whole doctrine. Those, therefore, who may wish for farther information on this abstruse but highly curious subject, are referred to the Miscellanea Analytica of Demoivre, Sterling's Method Differ., James Bernoulli de Seri. Infin, Simpson's Math Dissert., Waring's Medii. Analyt., Clark's translation of Lorgna's Series, the various works of Euler, and Lacroix Traite du Calcul Diff. et Int., where they will find nearly all the materials that have been hitherto collected respecting this branch of analysis.

†This mode of computation, which is one of the happiest and most useful discoveries of modern times, is due to Lord Napier, Baron of Merchision, in Scotland, who first published a table of these numbers, in the year 1614, under the title of Canon Mirificum Logarithmorum; which performance was eagerly received by the learned throughout Europe, whose efforts were immediately directed to the improvement and extensions of the new calculus, that had so unexpectedly presented itself.

Mr. Henry Briggs, in particular, who was, at that time, professor of geo metry in Gresham College, greatly contributed to the advancement of this doctrine, not only by the very advantageous alteration which he first introduced into the system of these numbers, by making 1 the logarithm of 10, instead of 2.3025852, as had been done by Napier; but also by the publication, in 1624 and 1633, of his two great works, the Arithmetica Logarithmica and the Trigonometria Britanica, both of which were formed upon the principle above mentioned: as are, likewise, all our common logarithmic tables, at present in use.

Or, when taken in a similar but more general sense. logarithms may be considered as the exponents of the powers to which a given or invariable number must be raised, in order to produce all the common, or natural numbers. Thus, if

ax=y, a*=y, a"y", &c. then will the indices x, x', x", &c. of the several powersof a, be the logarithms of the numbers y, y', y", &c. in the scale, or system, of which a is the base.

So that, from either of these formulæ it appears, that the logarithm of any number, taken separately, is the index of that power of some other number, which, when involved in the usual way, is equal to the given number.

And since the base a, in the above expressions, can be assumed of any value, greater or less than 1, it is plain that there may be an endless variety of systems of logarithms, answering to the same natural number.

It is, likewise, farther evident, from the first of these equations, that when y=1, x will be 0, whatever may be the value of a; and consequently the logarithm of 1 is always 0, in every system of logarithms.

And if x=1, it is manifest from the same equation that the base a will be y; which base is therefore the number whose proper logarithm, in the system to which it be.. longs, is 1.

Also, because a*=y, and ary, it follows from the multiplication of powers, that axa, or atyy; and, consequently, by the definition of logarithms, given above, x+x=log. yy, or

log. yy=log. y+log. y.

And, for a like reason, if any number of the equations ax=y, ay, ay', &c. be multiplied together, we shall have art'ta &c.=yyy' &c.; and consequently x+x+x" &c. log yyy" &c.; or

log. yyy" &c.; log. y+log. + log. y" &c.

".

See, for farther details on this part of the subject, the Introduction to my Treatise of Plane and Spherical Trigonometry, 8vo. 2d Edit. 1813; and for the construction and use of the tables consult those of Sherwin, Hutton, Tay. tor, Callet, and Borda, where every necessary information of this kind may be readily obtained..

From which it is evident, that the logarithm of the pro duct of any number of factors is equal to the sum of the logarithms of those factors.

Hence, if all the factors of a given number, in any case of this kind, be supposed equal to each other, and the sum of them be denoted by m, the preceding property will

then become

log. ymm log. y.

From which it appears, that the logarithm of the mth power of any number is equal to m times the logarithm of that number.

In like manner, if the equation ay be divided by a =y', we shall have, from the nature of powers, as before,

down in the first part of this article, x

log.log. y-log. y'.

y

Hence the logarithm of a fraction, or of the quotient arising from dividing one number by another, is equal to the logarithm of the numerator minus the logarithm of the denominator.

And if each member of the common equation a*=y be raised to the fractional power denoted by

m

n

176

we shall

And, consequently, by taking the logarithms, as before, m m x=log. yn, or log. y

m

log. J.

Where it appears, that the logarithm of a mixed root or power, of any number, is found by multiplying the lo garithm of the given number by the numerator of the index of that power, and dividing the result by the denominator.

And if the numerator m, of the fractional index, be in this case, taken equal to 1, the above formula will then become

From which it follows, that the logarithm of the nth root of any number is equal to the nth part of the logarithm of that number.

Hence, besides the use of logarithms, in abridging the operations of multiplication and division, they are equally applicable to the raising of powers and extracting of roots; which are performed by simply multiplying the given logarithin by the index of the power, or dividing it by the number denoting the root.

But although the properties here mentioned are common to every system of logarithms, it was necessary, for practical purposes, to select some one of them from the rest, and to adapt the logarithms of all the natural numbers to that particular scale.

And, as 10 is the base of our present system of arithmetic, the same number has accordingly been chosen for the base of the logarithmic system, now generally used.

So that, according to this scale, which is that of the common logarithmic tables, the numbers

10-4, 10-3, 10-2, 10-1, 10°, 101, 102, 103, 104, &c.

1, 10, 100, 1000, 10000, &c.

10000 1000'100'10'

have for their logarithms

-4, -3, -2, -1, 0, 1, 2, 3, 4, &c.

Which are evidently a set of numbers in arithmetical progression, answering to another set in geometrical progression; as is the case in every system of logarithms. And therefore, since the common or tabular logarithm of any number (n) is the index of that power of 10, which when involved, is equal to the given number, it is plain, from the following equation,

that the logarithms of all the intermediate numbers, in the above series, may be assigned by approximation, and made to occupy their proper places in the general scale.

It is also evident, that the logarithms of 1, 10, 100, 1000, &c. being 0, 1, 2, 3, &c. respectively, the logarithm of any number, falling between 0, and 1, will be 0 and some decimal parts; that of a number between 10 and 100, 1 and some decimal parts; of a number between 100 and 1000, 2 and some decimal parts; and so on, for other numbers of this kind.

1

And, for a similar reason, the logarithms of

1 1

of 10'100'

&c. or of their equals .1, .01, .001, &c. in the de1000' scending part of the scale, being —1, -2, -3, &c. the logarithm of any number, falling between 0 and 1, will be -1, and some positive decimal parts; that of a number between .1 and .01, -2, and some positive decimal parts; of a number between .01 and .001, -3, and some positive decimal parts; &c.

Hence, likewise, as the multiplying or dividing of any number by 10, 100, 1000, &c. is performed by barely increasing or diminishing the integral part of its logarithm by 1, 2, 3, &c. it is obvious that all numbers, which consist of the same figures, whether they be integral, fractional, or mixed, will have, for the decimal part of their logarithms, the same positive quantity.

So that, in this system, the integral part of any logarithm, which is usually called its index, or characteristic, is al ways less by 1 than the number of integers which the natural number consists of; and for decimals, it is the number which denotes the distance of the first significant figure from the place of units.

Thus, according to the logarithmic tables in common use, we have

Numbers.

Logarithms.